Problem: Solve for $x$ : $ 7|x + 4| - 2 = 4|x + 4| + 7 $
Solution: Subtract $ {4|x + 4|} $ from both sides: $ \begin{eqnarray} 7|x + 4| - 2 &=& 4|x + 4| + 7 \\ \\ { - 4|x + 4|} && { - 4|x + 4|} \\ \\ 3|x + 4| - 2 &=& 7 \end{eqnarray} $ Add ${2}$ to both sides: $ \begin{eqnarray} 3|x + 4| - 2 &=& 7 \\ \\ { + 2} &=& { + 2} \\ \\ 3|x + 4| &=& 9 \end{eqnarray} $ Divide both sides by ${3}$ $ \dfrac{3|x + 4|} {{3}} = \dfrac{9} {{3}} $ Simplify: $ |x + 4| = 3$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 4 = -3 $ or $ x + 4 = 3 $ Solve for the solution where $x + 4$ is negative: $ x + 4 = -3 $ Subtract ${4}$ from both sides: $ \begin{eqnarray} x + 4 &=& -3 \\ \\ {- 4} && {- 4} \\ \\ x &=& -3 - 4 \end{eqnarray} $ $ x = -7 $ Then calculate the solution where $x + 4$ is positive: $ x + 4 = 3 $ Subtract ${4}$ from both sides: $ \begin{eqnarray} x + 4 &=& 3 \\ \\ {- 4} && {- 4} \\ \\ x &=& 3 - 4 \end{eqnarray} $ $ x = -1 $ Thus, the correct answer is $x = -7 $ or $x = -1 $.